Simplify; express your answer in exponential form. Assume $r\neq 0, z\neq 0$. $\dfrac{{(r^{-4}z^{2})^{-4}}}{{(r^{5}z)^{2}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(r^{-4}z^{2})^{-4} = (r^{-4})^{-4}(z^{2})^{-4}}$ On the left, we have ${r^{-4}}$ to the exponent ${-4}$ . Now ${-4 \times -4 = 16}$ , so ${(r^{-4})^{-4} = r^{16}}$ Apply the ideas above to simplify the equation. $\dfrac{{(r^{-4}z^{2})^{-4}}}{{(r^{5}z)^{2}}} = \dfrac{{r^{16}z^{-8}}}{{r^{10}z^{2}}}$ Break up the equation by variable and simplify. $\dfrac{{r^{16}z^{-8}}}{{r^{10}z^{2}}} = \dfrac{{r^{16}}}{{r^{10}}} \cdot \dfrac{{z^{-8}}}{{z^{2}}} = r^{{16} - {10}} \cdot z^{{-8} - {2}} = r^{6}z^{-10}$